Is there any reason an intelligent item taking classes in intelligent item from the book Complete Eldritch Power could not also take ranks in other classes? Say fighter or wizard, so that they could have a familiar or take the leadership feat and have a follower to wield them?
Yes, intelligent magic items can multiclass if the short variant rule for multiclassing is used. (It's on page 55 of the Book of Eldritch Might III: The Nexus and page 69 of Complete Book of Eldritch Might.) Those rules also include suggestions for how to handle intelligent magic items casting spells with somatic components.
There isn't any reason why they cannot...
...so long as they can meet all the prerequisites (if any) and requirements of the class.
Note that levels of fighter will not help your combat potential much, unless the item in question can self wield. Nor will a follower-wielder gain your attack bonus from fighter levels.
And casting spells as a wizard would be rather problematic without the ability to gesture for the somatic components, not to mention handling the spell components.
I would suggest considering psionics for ease of use, otherwise you'll have to be really careful with your level choices.
Note that all magic items are considered constructs, (DMG p.268, paragraph 2 under the Intelligent items section) and thus could always take levels in Construct, gaining feats, skill points, and HD/HP, etc..